Papers › Homology representations of compactified configurations on graphs applied to ℳ_(2,n)
Homology representations of compactified configurations on graphs applied to ℳ_(2,n)
Christin Bibby, Melody Chan, Nir Gadish, Claudia He Yun
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We obtain new calculations of the top weight rational cohomology of the moduli spaces ℳ_(2,n), equivalently the rational homology of the tropical moduli spaces Δ_(2,n), as a representation of Sₙ. These calculations are achieved fully for all n≤10, and partially -- for specific irreducible representations of Sₙ -- for n≤22. We also present conjectures, verified up to n=22, for the multiplicities of the irreducible representations stdₙ and stdₙ⊗sgnₙ. We achieve our calculations via a comparison with the homology of compactified configuration spaces of graphs. These homology groups are equipped with commuting actions of a symmetric group and the outer automorphism group of a free group. In this paper, we construct an efficient free resolution for these homology representations, from which we extract calculations on irreducible representations one at a time, simplifying the calculation of these homology representations.
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